POP Implementation of Mesoscale eddy diffusivity based on mixing theory

Here are my notes on the development of a parameterization for eddy diffusivity. The scheme is based on the paper by Michael Bates, Ross Tulloch, John Marshall, and Raffaele Ferrari. I'm working with Gokhan Danabasoglu and Matt Long here at NCAR and John Marshall (co-author) at MIT.

The general approach of this effort will be to follow John and Gokhan's implementation notes, isolating each of the terms contributing to the eddy diffusivity given by equation 6 in Bates et al. and comparing the CESM implementation of those terms with Bates, reanalysis, or other observations.

Implementation Notes on Diffusivity with Steering Level Suppression (Gokhan and John Marshall)

12 November 2014

Starting with

$\quad\quad\quad K=u_{rms}∗L_{mix}$

the general form for diffusivity, $K$, is given by equation (6) of Bates et al. (2014). Namely,

$\quad\quad\quad L_{mix} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} $

Here,

$u_{rms}$ is the root-mean-square (rms) eddy velocity;

$L_{mix}$ is the mixing length;

$L_{eddy}$ is the eddy diameter (depth independent);

$u_{mean}$ is the mean zonal velocity (resolved);

$c$ is the zonal eddy phase speed (depth independent);

$\Gamma = 0.35$

$b1 = 4$

$\quad\quad\quad\quad\quad\quad$ Eddy Length Scales:

Rossby deformation radius = $L_r = {c_r \over |f|}$

Equatorial Rossby deformation radius $= L_{req} = {\sqrt{c_r \over 2\beta}}$

Rhines scale $= L_{Rh} = {\sqrt{u_{rms} \over \beta}} \sim {\sigma_{vi} \over \beta}$

$c_r$ is the first baroclinic wave speed computed following equation (2.2) of Chelton et al. (1998) with $m=1$;

$f$ is the Coriolis parameter;

$\beta$ is the latitudinal variation of the Coriolis parameter; and

$\sigma_{vi}$ is the Eady growth rate given by

$\sigma_{vi} = {f \over \sqrt{R_i}}$

with $R_i$ the vertically integrated (over the 100 – 2000 m depth range) Richardson number.

So, any one these length scales could be used as an eddy length scale. An alternative is

$L_{eddy} = min (L_r, L_{req}, L_{Rh}).$

$\quad\quad\quad\quad\quad\quad$ Eddy Velocity:

${u_{rms} = alpha*\sigma_{vi}*L_r}$

where $\sigma_{vi}$ is the Eady growth rate based on local Richardson number and $\alpha$ is a scaling constant.

$\quad\quad\quad\quad\quad\quad$ Zonal phase speed:

$c = - \beta * L_r^2$

References:

Bates, M., R. Tulloch, J. Marshall, and R. Ferrari, 2014: Rationalizing the spatial distribution of mesoscale eddy diffusivity in terms of mixing length theory. J. Phys. Oceanogr., 44, 1523-1540, doi: 10.1175/JPO-D-13-0130.1.

Chelton, D. B., R. A. deSzoeke, M. G. Schlax, K. E. Naggar, and N. Siwertz, 1998: Geographical variability of the first baroclinic Rossby radius of deformation. J. Phys. Oceanogr., 28, 433-460.

Tullock, R., J. Marshall, and K. S. Smith, 2009: Interpretation of the propagation of surface altimetric observations in terms of planetary waves and geostropic turbulence. J. Geophys. Res., 114, C02005, doi: 10.1029/2008JC005055.

Questions:

In the 2D implementation, $u_{rms}$ and $u_{mean}$ specifications: upper-ocean vertically or integrated or at $z = 0$?

A: $U_{rms}$ is not depth dependent; both $u_{rms}$ and $u_{mean}$ are for surface only

In the 2D implementation, vertical profile will be specified by $N2(z)$?

A: Yes, $N^2(z) \over N_{ref}(z)$ to be more precise

Local $R_i$ use imbedded in sigma in $u_{rms}$ calculation?

A: No, $u_{rms}$ is depth independent

alpha = ?

A: trial and error

Cancellation of $f$’s in $u_{rms}$ calculation?

Zonal phase speed equation correct? Both $\beta$ and $L_r$ will be positive, >producing $c < 0$ always. This appears to be in contrast with Tullock et al. (2009).

A: What is plotted in Bates is (U-c)

$\color{green}{\quad CESM \quad Implementation \quad of \quad Steering \quad Parameterization}$

During the development of the parameterization several tuning modifications were made to the terms listed above to better match the observations.

New implementation of Eady Growth rate (sigma) that avoids undefined values at the equator. See alternate derivation at end of this page.

$\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}$ is replace by $\sigma_{vi} = \quad {{\require\cancel f m^2} \over \cancel f N}$

Don't use Rhine Scale in calculation of $L_{eddy}$

$\quad\quad\quad L_{eddy} = min (L_r, L_{req}, \cancel{L_{Rh}})$

Eddy phase speed calculation modified to avoid undefined value at the equator and limited to better match Bates paper.

$\quad\quad\quad \cancel{c = - \beta * {L_r^2}}$ is replace with $c = max(- \beta * L_{eddy}^2,-20)$

$u_{rms}$ is hard to capture in this implememtation due to the coarse model resolution and use of mean vs resolved states. We imposed a $5 cm/sec$ minimum on $U_{rms}$ to tune to surface obs of Bates and included a scaling constant $\alpha$ to handle the bias introduced by using mean state instead of resolved state.

$\quad\quad\quad u_{rms} = max(5.,alpha*\sigma_{vi}*L_r)$

$\quad\quad\quad \alpha=4$ The scaling constant for $U_{rms}$

The Eddy diffusivity calculated by our implementation was weak so be bumped up $\Gamma$ from .35 to 1.75 (5x increase)

$\quad\quad\quad \Gamma=.35$ replated with $\Gamma = 1.75$

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Installed watermark.py. To use it, type:
  %load_ext watermark
The watermark extension is already loaded. To reload it, use:
  %reload_ext watermark
Last updated: 02/06/2015 16:56:19 MDT

CPython 2.7.9
IPython 2.2.0
Populating the interactive namespace from numpy and matplotlib
/loan/jet/steering/g.e11.GIAF.T62_gx1v6.steer.018.pop.h.nday1.290-310.nc

WARNING: pylab import has clobbered these variables: ['norm', 'num2date', 'deg2rad', 'f', 'yticks', 'gamma']
`%matplotlib` prevents importing * from pylab and numpy

$\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} $

NOTES: >$K=u_{rms}∗{\Gamma * \color{red}{L_{eddy}} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} $ >The Eddy Length scale is parameterized as $L_{eddy} = min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}}).$ > >Since the two length scales we are using in the parameterization of $L_{eddy}$ depend on the Baroclinic wave speed $c_r$ we will first check $c_r$ against Chelton 1998.

$\quad\quad\quad\quad\quad\quad\quad$ First Baroclinic Wave Speed $c_r$ CESM

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$\quad\quad\quad\quad\quad\quad$ Chelton Baroclinic gravity wave phase speed

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The barclinic wave speed looks reasonable so lets compare the Paramaterized Rossby Radius to Chelton

$\quad\quad\quad\quad\quad\quad$ First Baroclinic Rossby Radius CESM $\quad L_{eddy}= min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}}).$

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# <!-- collapse=True --> 

fcor = nc.variables['FCOR'][0,:,:]
fcor_nearest = pyresample.kd_tree.resample_nearest(orig_def, fcor, \
        targ_def25, radius_of_influence=500000, fill_value=None)
absf=np.abs(fcor_nearest)
l_rossby_calc=c_rossby_nearest/absf

btp = nc.variables['BTP'][0,:,:]
btp_nearest = pyresample.kd_tree.resample_nearest(orig_def, btp, \
        targ_def25, radius_of_influence=500000, fill_value=None)

l_rossbyeq_calc=np.sqrt(c_rossby_nearest/(2*btp_nearest))
l_eddy_calc=np.minimum(l_rossby_calc,l_rossbyeq_calc)

### convert regridded radius from cm to km
cm2km=1.e-5
l_eddy_calc_km=l_eddy_calc*cm2km
clevs=[10,20,30,40,50,60,80,100,150,230]
fig=MapZoneContour(targ_def25.lons,targ_def25.lats,l_eddy_calc_km,figsize=(14,9),
                    levels=clevs,
                    llcrnrlon=25.,
                    title="CESM First Baroclinic Rossby Radius (km) $L_{eddy} \sim min(l_r,l_{req})$",
                    fmt='%.1i',
                    imagename='Fig_2_CESM_First_Baroclinic_Rossby_Radius.png')
lrhines = nc.variables['L_RHINES'][0,:,:]
lrhines_nearest = pyresample.kd_tree.resample_nearest(orig_def, lrhines, \
        targ_def25, radius_of_influence=500000, fill_value=None)

###clevs=np.logspace(1,15,40)
###norm=LogNorm()
###fig,ax,cbar =MapContourg(targ_def0,lrhines_nearest,
###                    addzonal=True,
###                    levels1=clevs,
###                    norm=norm,
###                    figsize=(14,5),
###                    title="$L_{rhines} (cmn^2)$ log scale")
###cbar.set_ticks(np.logspace(1,15,15))
if False:
    l_eddy1=np.maximum(l_eddy_calc,lrhines_nearest)
    l_eddy1_km=l_eddy1*cm2km
    clevs=[10,20,30,40,50,60,80,100,150,230]
    fig2=MapZoneContour(targ_def25.lons,targ_def25.lats,l_eddy1_km,figsize=(14,9),
                    levels=clevs,
                    llcrnrlon=25.,
                    title="CESM First Baroclinic Rossby Radius (km) using rhines scale $L_{eddy} \sim max(l_rhines,min(l_r,l_{req}))$",
                    fmt='%.1i' )

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$\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})$

NOTES:

\[K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - \color{red}{c}|^2 /u_{rms}^2 (z=0)}\]

$\require{cancel}\cancel{c = - \beta * {L_r^2}}$ $L_r$ too high at equator

$c = - \beta * L_{eddy}^2$

$\quad\quad\quad\quad\quad$ Eddy Phase speed CESM

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# <!-- collapse=True --> 
if False:
    #beta
    clevs=arange(1.e-14,25.e-14,1e-14)
    fig,ax,cbar =MapContourg(targ_def0,btp_nearest,
                    levels1=clevs,
#                    norm=norm,
                    addzonal=True,
                    figsize=(14,5),
                    title="Beta (1/cm*s)",
                    imagename='Fig_3_Beta.png')
l_eddy_calc_sq=l_eddy_calc*l_eddy_calc

if False:
    clevs=np.logspace(1,15,40)
    norm=LogNorm()
    fig,ax,cbar =MapContourg(targ_def0,l_eddy_calc_sq,
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    title="$L_{eddy}^2 (cm^2)$ log scale")
    #                    imagename='Fig_4_L_eddy_SQ.png')

    #cbar.ax.get_xaxis().get_major_formatter().labelOnlyBase = True
    #cbar.ax.get_yaxis().set_major_formatter(plt.LogFormatter(10,  labelOnlyBase=True))
    #ax.yaxis.set_major_locator(ticker.MultipleLocator)
    #ax.yaxis.set_major_formatter(LogFormatter())
    #cbar.formatter.set_scientific(False)
    #cbar.formatter.set_powerlimits((0, 0))
    #cbar.update_ticks()
    cbar.set_ticks(np.logspace(1,15,15))
    fig.savefig('Fig_4_L_eddy_SQ.png',bbox_inches='tight',dpi=200)


c=btp_nearest*l_eddy_calc*l_eddy_calc # units=1/(cm s) * cm^2 = cm/s


clevs=arange(-10,20,1)
fig,ax,cbar = MapContourg(targ_def0,c,
                    addzonal=True,
                    levels1=clevs,
#                    norm=norm,
                    setover='darkred',
                    figsize=(14,5),
                    title="eddy phase speed c = beta*$l_{eddy}^2$ (cm/s) linear scale(NOTE I'm plotting positive c here - changed to negative after this plot")
cbar.set_ticks(np.linspace(-10,20,7))
fig.savefig('Fig_5_eddy_phase_speed.png',bbox_inches='tight',dpi=200)

if False:
    c_rhines=btp_nearest*l_eddy1*l_eddy1 
    clevs=arange(-10,20,1)
    fig,ax,cbar = MapContourg(targ_def0,c_rhines,
                    addzonal=True,
                    levels1=clevs,
#                    norm=norm,
                    setover='darkred',
                    figsize=(14,5),
                    title="eddy phase speed including rhines scale c = beta*$l_{eddy}^2$ (cm/s) linear scale(NOTE I'm plotting positive c here - changed to negative after this plot")
    cbar.set_ticks(np.linspace(-10,20,7))

c=-1.*c

Compared to Hughes, the phase speed is much too high near the equator.

The first tuning mod is to limit the phase speed to 20cm/s.

$\quad \quad \quad \quad $Eddy Phase speed CESM $(c = max(- \beta * L_{eddy}^2,-20) )$

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$\quad\quad\quad\quad\quad\quad$ Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09

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$\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})$

NOTES:

$K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |\color{red}{u_{mean}} - c|^2 /u_{rms}^2 (z=0))} $

Here $U_{mean}$ is just CESM surface velocity.

$\quad\quad \quad \quad\quad $ Zonal Velocity CESM

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# <!-- collapse=True --> 
#################  PLOT (u-c)^2  convert m^2/s^2 #######################
umean=nc.variables['U_MEAN'][0,:,:]
umean10_nearest = pyresample.kd_tree.resample_nearest(orig_def, umean,
                targ_def0, radius_of_influence=500000, fill_value=None)

umean10_mps=umean10_nearest*1.e-2

clevs=np.linspace(-60,60,40)
fig,ax,cbar=MapContourg(targ_def0,umean10_nearest,
                    addzonal=True,
                    figsize=(14,5),
                    levels1=clevs,
#                    setover='darkred',
#                    setunder='darkblue',
                    title="U zonal velocity (cm/s)")
#fig.subplots_adjust(left=0.15, right=.9, bottom=0.1, top=0.9);
cbar.set_ticks(arange(-60,60,10))
pylab.xlim([-20.,20.]);
fig.savefig('Fig_7_POP_UVEL.png',bbox_inches='tight',dpi=200)

if False:
    
    # U averaged over 0-50m
    
    clevs=np.linspace(-60,60,40)
    fig,ax,cbar=MapContourg(targ_def0,u50_nearest,
                    addzonal=True,
                    figsize=(14,5),
                    levels1=clevs,
#                    setover='darkred',
#                    setunder='darkblue',
                    title="U 0-50m avg zonal velocity (cm/s)")
    #fig.subplots_adjust(left=0.15, right=.9, bottom=0.1, top=0.9);
    cbar.set_ticks(arange(-60,60,10))
    pylab.xlim([-20.,20.]);




# get 2D versions of the lat and lon variables add longitude start point here!
lon_ecco_orig, lat_ecco_orig = np.meshgrid(lon_ecco[:], lat_ecco[:])

orig_ecco_def = pyresample.geometry.GridDefinition(lons=lon_ecco_orig, lats=lat_ecco_orig)

ecco_u_nearest = pyresample.kd_tree.resample_nearest(orig_ecco_def, ecco_u, \
        targ_def0, radius_of_influence=500000, fill_value=None)
ecco_u_nearest_masked=np.ma.masked_where(ecco_u_nearest<-10, ecco_u_nearest, copy=True)

#print ecco_u_cyc_masked[:,0]

clevs=np.linspace(-60,60,40)
#ecco_u_nearest = pyresample.kd_tree.resample_nearest(orig_def, ecco_u, \
#        targ_def, radius_of_influence=500000, fill_value=None)
fig,ax,cbar=MapContourg(targ_def0,ecco_u_nearest_masked*100,
                    addzonal=True,
                    figsize=(14,5),
                    levels1=clevs,
#                    extend='both',
                    title="ECCO U zonal velocity (cm/s)")
cbar.set_ticks(arange(-60,60,10))
pylab.xlim([-20.,20.]);
fig.savefig('Fig_8_ECCO_UVEL.png',bbox_inches='tight',dpi=200)

$\color{green}{(U-c)\quad Term}$

NOTES: $c = max(- \beta * L_r^2,-20)$ )

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (U-c) CESM

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$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (U-c) Bates et al

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$\color{green}{(U-c)^2 \quad Term}$

NOTES: $c = max(- \beta * L_{eddy}^2,-20)$

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2$ CESM

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# <!-- collapse=True --> 

uminusc2sq = nc.variables['U_MINUS_C_SQ'][0,0,:,:]
uminusc2sq_nearest = pyresample.kd_tree.resample_nearest(orig_def, uminusc2sq, \
        targ_def0, radius_of_influence=500000, fill_value=None)
uminusc2sq_mps=uminusc2sq_nearest*1.e-4

if False:
    clevs=np.logspace(-5,0,40)
#    norm=matplotlib.colors.SymLogNorm(linthresh=1e-4, vmin=-1000, vmax=-.001)
    norm=LogNorm()
    fig,ax,cbar=MapContourg(targ_def0,uminusc2sq_mps,
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    title=" ${|(U-c)|}^2$ using new $c_{eddy} {(m^2/s^2)}$ max = %g/min = %g "%(uminusc2sq_mps.max(),uminusc2sq_mps.min()))

    # Customize y tick lables
    yticks=[1e-5,3e-5,1e-4,3e-4,1e-3,3e-3,1e-2,3e-2,1e-1,3e-1,1]
    cbar.set_ticks(yticks)
    cbar.set_ticklabels(yticks)
    pylab.xlim([0.,.1])
    fig.savefig('Fig_12_U_minus_c_SQ_new_log.png',bbox_inches='tight',dpi=200)

clevs=np.linspace(0,.06,25)
fig,ax,cbar=MapContourg(targ_def0,uminusc2sq_mps,
                    addzonal=True,
                    levels1=clevs,
                    setover='darkred',
                    setunder='black',
                    figsize=(14,5),
                    title=" ${|(U-c)|}^2$ using new $c_{eddy} {(m^2/s^2)}$ max = %g/min = %g "%(uminusc2sq_mps.max(),uminusc2sq_mps.min()))
#customize colorbar ticks
yticks=[0,.02,.04,.06]
cbar.set_ticks(yticks)
#customize zonal ticks
pylab.xlim([0,.06])
pylab.xticks([0,.03,.06]);
fig.savefig('Fig_13_U_minus_c_SQ_new.png',bbox_inches='tight',dpi=200)


uminusc2_calc_sq_mps=uminusc2*uminusc2*1.e-4
if False:
    clevs=np.linspace(0,.06,25)
    fig,ax,cbar=MapContourg(targ_def0,uminusc2_calc_sq_mps,
                    addzonal=True,
                    levels1=clevs,
                    setover='darkred',
                    setunder='black',
                    figsize=(14,5),
                    title=" ${|(U-c)|}^2 {(m^2/s^2)}$ calc max = %g/min = %g "%(uminusc2_calc_sq_mps.max(),uminusc2_calc_sq_mps.min()))
    #customize colorbar ticks
    yticks=[0,.02,.04,.06]
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.06])
    pylab.xticks([0,.03,.06]);

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2$ Bates et al

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$\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma}$

NOTES: >The original derivation of $\sigma_{vi}$ is undefined at the equator because of $f$

$\sigma_{vi} = {f \over \sqrt{R_i}}$

The new derivation of $\sigma_{vi}$:

$ R_i = {f^2 N^2 _}= $

$\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}$

$\quad\quad\quad\quad\quad $Eady Growth Rate $(\sigma) \quad$ CESM

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$\quad\quad\quad\quad\quad $Eady Growth Rate $(\sigma) \quad$ Bates et al

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$\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}$

NOTES: >$K=\color{red}{u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} $ > >$u_{rms} = alpha*{\sigma_{vi}}*L_{eddy}$ > >alpha (scaling constant) = 4 > >$\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}$ > >$u_{rms}$ is limited to 5 cm/s $ max(u_{rms},5.)$

$\quad\quad\quad\quad\quad\quad\quad$ Eddy Velocity $(u_{rms})\quad$ CESM

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# <!-- collapse=True --> 
alpha=4
cm2m=.01
#
urms_avg  =  nc.variables['URMS_AVG'][0,:,:]
urms_avg_nearest = pyresample.kd_tree.resample_nearest(orig_def, urms_avg, 
        targ_def0, radius_of_influence=500000, fill_value=None)
urms_avg_mps=urms_avg_nearest*cm2m

if False:
    
    clevs=np.logspace(-3,1,40)
    norm=LogNorm()
    fig,ax,cbar=MapContourg(targ_def0,urms_avg_mps,                
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    setover='darkred',
                    setunder='darkblue',
                    title="$u_{rms} ({m \over s})$ using old sigma log plot(alpha= %.1f,max = %.1f/min = %.2G) "%(alpha,urms_avg_mps.max(),urms_avg_mps.min()))
    #customize colorbar ticks
    yticks=np.logspace(-5,1,7)
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.22])
    pylab.xticks([0,.1,.2]);

    
    
    clevs=np.linspace(0,.4,40)
    fig,ax,cbar=MapContourg(targ_def0,urms_avg_mps,
                    addzonal=True,
                    levels1=clevs,
                    figsize=(14,5),
                    title="$u_{rms} ({m \over s})$ using old sigma  (alpha=4,max = %.1f/min = %.1f) "%(urms_avg_mps.max(),urms_avg_mps.min()))
    #customize colorbar ticks
    yticks=[0,.1,.2,.3,.4]
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.22])
    pylab.xticks([0,.1,.2]);




urms_avg1 = nc.variables['URMS_AVG1'][0,0,:,:]
urms_avg1_nearest = pyresample.kd_tree.resample_nearest(orig_def, urms_avg1, 
        targ_def0, radius_of_influence=500000, fill_value=None)
urms_avg1_mps=urms_avg1_nearest*cm2m

if False:
    clevs=np.logspace(-3,1,40)
    norm=LogNorm()
    fig,ax,cbar=MapContourg(targ_def0,urms_avg1_mps,                
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    setover='darkred',
                    setunder='darkblue',
                    title="$u_{rms} ({m \over s})$ using new sigma log plot(alpha= %.1f,max = %.1f/min = %.2G) "%(alpha,urms_avg1_mps.max(),urms_avg1_mps.min()))
    #customize colorbar ticks
    yticks=np.logspace(-5,1,7)
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.22])
    pylab.xticks([0,.1,.2]);
    fig.savefig('Fig_16_Urms_new_log.png',bbox_inches='tight',dpi=200)

clevs=np.linspace(0,.4,40)
fig,ax,cbar=MapContourg(targ_def0,urms_avg1_mps,
                    addzonal=True,
                    levels1=clevs,
                    figsize=(14,5),
                    title="$u_{rms} ({m \over s})$ using new sigma (alpha= %.1f,max = %.1f/min = %.2G) "%(alpha,urms_avg1_mps.max(),urms_avg1_mps.min()))
#customize colorbar ticks
yticks=[0,.1,.2,.3,.4]
cbar.set_ticks(yticks)
#customize zonal ticks
pylab.xlim([0,.22])
pylab.xticks([0,.1,.2]);
fig.savefig('Fig_17_Urms_new.png',bbox_inches='tight',dpi=200)

cm2m=.01
l_eddy_calc_m=l_eddy_calc*cm2m
urms_calc_mps=alpha*sigma_avg1_nearest*l_eddy_calc_m
if False:
    clevs=np.linspace(0,.4,40)
    fig,ax,cbar=MapContourg(targ_def0,urms_calc_mps,
                    addzonal=True,
                    levels1=clevs,
                    figsize=(14,5),
                    title="$u_{rms} ({m \over s})$ calc value  (alpha= %.1f,max = %.1f/min = %.1f) "%(alpha,urms_calc_mps.max(),urms_calc_mps.min()))
    #customize colorbar ticks
    yticks=[0,.1,.2,.3,.4]
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.22])
    pylab.xticks([0,.1,.2]);

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} $ Bates et al

In [51]:

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$\color{green}{u_{rms}^2 \quad Term} \quad$

NOTES: >$K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /\color{red}{u_{rms}^2} (z=0)} $ > >$u_{rms}^2$ is a surface value

$\quad\quad\quad\quad\quad\quad\quad$ Eddy Velocity Squared $(u_{rms}^2)\quad$ CESM

In [52]:

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#### <!-- collapse=True --> 
urms_avg1_sq = nc.variables['URMS_AVG1_SQ'][0,0,:,:]
urms_avg1_sq_nearest = pyresample.kd_tree.resample_nearest(orig_def, urms_avg1_sq, 
        targ_def0, radius_of_influence=500000, fill_value=None)
urms_avg1_sq_mps=urms_avg1_sq_nearest*1.e-4
###urms_avg1_sq_mps =urms_avg1_mps*urms_avg1_mps

urms_avg_sq = nc.variables['URMS_AVG_SQ'][0,:,:]
urms_avg_sq_nearest = pyresample.kd_tree.resample_nearest(orig_def, urms_avg_sq, 
        targ_def0, radius_of_influence=500000, fill_value=None)
urms_avg_sq_mps=urms_avg_sq_nearest*1.e-4

if False:
    clevs=np.logspace(-5,1,42)
    norm=LogNorm()
    fig,ax,cbar=MapContourg(targ_def0,urms_avg_sq_mps,                
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    setover='darkred',
                    setunder='darkblue',
                    title="$u_{rms}^2 old sigma({m^2 \over s^2})$ log plot (alpha=%.1f,max = %.2G/min = %.2G) "%(alpha,urms_avg_sq_mps.max(),urms_avg_sq_mps.min()))
    #customize colorbar ticks
    yticks=np.logspace(-5,1,7)
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.06])
    pylab.xticks([0,.03,.06]);



    ##clevs=np.linspace(0,.06,40)
    clevs=np.linspace(.001,.06,40)
    fig,ax,cbar=MapContourg(targ_def0,urms_avg_sq_mps,
                    addzonal=True,
                    levels1=clevs,
                    setover='darkred',
                    setunder='darkblue',
                    figsize=(14,5),
                    title="$u_{rms}^2 old sigma ({m^2 \over s^2})$ (alpha=%.1f,max = %.2G/min = %.2G) "%(alpha,urms_avg_sq_mps.max(),urms_avg_sq_mps.min()))
    #customize colorbar ticks
    yticks=[.001,.02,.04,.06]
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.06])
    pylab.xticks([0,.03,.06]);


    
    
    clevs=np.logspace(-5,1,42)
    norm=LogNorm()
    fig,ax,cbar=MapContourg(targ_def0,urms_avg1_sq_mps,                
                    addzonal=True,
                    levels1=clevs,
                    norm=norm,
                    figsize=(14,5),
                    setover='darkred',
                    setunder='darkblue',
                    title="$u_{rms}^2 new sigma({m^2 \over s^2})$ log plot (alpha=%.1f,max = %.2G/min = %.2G) "%(alpha,urms_avg1_sq_mps.max(),urms_avg1_sq_mps.min()))
    #customize colorbar ticks
    yticks=np.logspace(-5,1,7)
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.06])
    pylab.xticks([0,.03,.06]);
    fig.savefig('Fig_18_Urms_SQ_log.png',bbox_inches='tight',dpi=200)



##clevs=np.linspace(0,.06,40)
clevs=np.linspace(.001,.06,40)
fig,ax,cbar=MapContourg(targ_def0,urms_avg1_sq_mps,
                    addzonal=True,
                    levels1=clevs,
                    setover='darkred',
                    setunder='darkblue',
                    figsize=(14,5),
                    title="$u_{rms}^2 new sigma ({m^2 \over s^2})$ (alpha=%.1f,max = %.2G/min = %.2G) "%(alpha,urms_avg1_sq_mps.max(),urms_avg1_sq_mps.min()))
#customize colorbar ticks
yticks=[.001,.02,.04,.06]
cbar.set_ticks(yticks)
#customize zonal ticks
pylab.xlim([0,.06])
pylab.xticks([0,.03,.06]);
fig.savefig('Fig_19_Urms_SQ.png',bbox_inches='tight',dpi=200)

urms_calc_mps_sq=urms_calc_mps*urms_calc_mps

if False:
    clevs=np.linspace(.001,.06,40)
    fig,ax,cbar=MapContourg(targ_def0,urms_calc_mps_sq,
                    addzonal=True,
                    levels1=clevs,
                    setover='darkred',
                    setunder='darkblue',
                    figsize=(14,5),
                    title="$u_{rms}^2 ({m^2 \over s^2})$ (alpha=%.1f,max = %.2G/min = %.2G) "%(alpha,urms_calc_mps_sq.max(),urms_calc_mps_sq.min()))
    #customize colorbar ticks
    yticks=[.001,.02,.04,.06]
    cbar.set_ticks(yticks)
    #customize zonal ticks
    pylab.xlim([0,.06])
    pylab.xticks([0,.03,.06]);

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 $ Bates et al

In [53]:

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$\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}$

NOTES: >$K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * \color{red}{|u_{mean} - c|^2 /{u_{rms}^2} (z=0)}} $ > > >$c = max(- \beta * L_{eddy}^2,-20)$ > >$u_{rms} = max(u_{rms},5.)$ > >$u_{rms}^2$ is a surface value

$\quad\quad\quad\quad\quad\quad$ ${(U-c)}^2 \over {u_{rms}^2} \quad$ CESM

In [54]:

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$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} $ Bates et al

In [55]:

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$\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}$

NOTES: > >$c = max(- \beta * L_{eddy}^2,-20)$ > >$u_{rms} = max(u_{rms},5.)$ > >$u_{rms}^2$ is a surface value > >$b1$ (scaling constant) = 4.

$\quad\quad\quad\quad\quad\quad\quad$ Suppression Factor CESM

In [56]:

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$\quad\quad\quad\quad\quad\quad\quad$ Suppression Factor Bates

In [57]:

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$\quad\quad\quad\quad\quad\quad\quad$ Suppression Factor (Zonal x depth) CESM

In [58]:

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clevs=np.linspace(0,1,11)

z_t= ncregrid.variables['z_t'][:]

latregrid=ncregrid.variables['lat'][:]
lonregrid=ncregrid.variables['lon'][:]
lon=lonregrid

suppregrid= ncregrid.variables['SUPP'][0,:,:,:]
suppregridz=suppregrid.mean(axis=2)
clevs=np.linspace(0,1,100,endpoint=True)
yticks=[0,.2,.4,.6,.8,1.]
fig,ax,cbar=plotzonalocn(latregrid, z_t, suppregridz,
                    levels1=clevs,
                    levels2=clevs,
#                    equalcontourscale=True,
                    figsize=(10.5,6),
                    title="Suppression factor (custom contours max = %.1f/min = %.1f) "%(suppregridz.max(),suppregridz.min()))

# Add axis labels and plot title                                                                                                                                                                                              
plt.xlabel('Latitude')                                                                                                                                                                                                
plt.ylabel('Depth (cm)')                                                                                                                                                                                              

# Customize y tick lables                                                                                                                                                                                                                                                                                                                                                                                                              
cbar.set_ticks(yticks)                                                                                                                                                                                                
cbar.set_ticklabels(yticks)                                                                                                                                                                                           
cbar.set_label('fraction')                                                                                                                                                                                            
fig.savefig('Fig_23_Suppression_zonal_depth.png',bbox_inches='tight',dpi=200)

$\quad\quad\quad\quad\quad\quad\quad$ Suppression Factor (Zonal x depth) Bates

In [59]:

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$\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}$

NOTES: >$\color{red}{L_{mix}} = \Gamma * L_{eddy} * Suppression$ > >$Suppression= {1 \over (1 + b1 * |u_{mean} - c|^2 /u_{rms (z=0)}^2 )}$ > >$\color{red}{L_{mix}} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}$ > >$\Gamma = 1.75$ (Tuning mod: the original Gamma of .35 produced a Kappa with the correct structure but too weak.;

$\quad\quad\quad\quad\quad\quad\quad$ LMIX CESM

In [60]:

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$\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})$

NOTES: >###$\color{red}{K}=u_{rms}∗L_{mix}$

$\quad\quad\quad\quad\quad\quad\quad$ Eddy Diffusivity (K) CESM

In [61]:

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$\quad\quad\quad\quad\quad\quad\quad$ Eddy Diffusivity (K) Bates

In [62]:

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$\quad\quad\quad\quad\quad\quad\quad$ (Zonal x Depth) CESM

$\quad\quad\quad\quad\quad\quad\quad\quad\quad$ K limited to (100 < K < 10000)

In [63]:

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# <!-- collapse=True --> 

###ncregrid = Dataset(basedir+'/steering/g.e11.GIAF.T62_gx1v6.steer.017.regrid.nc')
z_t= ncregrid.variables['z_t'][:]

latregrid=ncregrid.variables['lat'][:]
lonregrid=ncregrid.variables['lon'][:]
lon=lonregrid

kvert= ncregrid.variables['KVERT'][0,:,:,:]
kvertz=kvert.mean(axis=2)/1.e4

clevs=np.arange(0,100,20)
clevs=np.append(clevs,np.arange(100,1000,100))
clevs=np.append(clevs,np.arange(1000,4000,1000))
fig,ax,cbar=plotzonalocn(latregrid, z_t, kvertz,
                    levels1=clevs,
                    levels2=clevs,
                    equalcontourscale=True,
                    figsize=(10.5,6),
                    title="Eddy diffusivity $K$ (custom contours max = %.1f/min = %.1f) "%(kvertz.max(),kvertz.min()))

# Add axis labels and plot title                                                                                                                                                                                              
plt.xlabel('Latitude')                                                                                                                                                                                                
plt.ylabel('Depth (cm)')                                                                                                                                                                                              

# Customize y tick lables                                                                                                                                                                                                     
yticks=clevs                                                                                                                                                                                                          
cbar.set_ticks(yticks)                                                                                                                                                                                                
cbar.set_ticklabels(yticks)                                                                                                                                                                                           
cbar.set_label('m**2/s')                                                                                                                                                                                            
fig.savefig('Fig_29_K_zonal_depth_custom.png',bbox_inches='tight',dpi=200)

clevs=np.linspace(0,2500,100,endpoint=True)
yticks=[0,500,1000,1500,2000,2500]
fig,ax,cbar=plotzonalocn(latregrid, z_t, kvertz,
                    levels1=clevs,
                    levels2=clevs,
#                    equalcontourscale=True,
                    figsize=(10.5,6),
                    title="Eddy diffusivity $K$ (Linear Scale max = %.1f/min = %.1f) "%(kvertz.max(),kvertz.min()))

# Add axis labels and plot title                            
levs=[500,1000,1500]
cs=ax.contour(latregrid, z_t, kvertz,levels=levs,colors='w',linewidths=2)
plt.xlabel('Latitude')                                                                                                                                                                                                
plt.ylabel('Depth (cm)')                                                                                                                                                                                              
plt.clabel(cs, inline=1, fontsize=15)
# Customize y tick lables                                                                                                                                                                                                                                                                                                                                                                                                            
cbar.set_ticks(yticks)                                                                                                                                                                                                
cbar.set_ticklabels(yticks)                                                                                                                                                                                           
cbar.set_label('m**2/s')                                                                                                                                                                                            
fig.savefig('Fig_30_K_zonal_depth_linear.png',bbox_inches='tight',dpi=200)

Here is the Bate's version

In [64]:

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$\quad\quad\quad\quad\quad\quad\quad$ Zonal average of the N2 normalized scaling CESM

In [65]:

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clevs=np.linspace(0,1,11)
###ncregrid = Dataset(basedir+'/steering/g.e11.GIAF.T62_gx1v6.steer.015.regrid.nc')
n2norm=ncregrid.variables['BFREQ_SQ_NORM'][0,:,:,:]
n2normz=n2norm.mean(axis=2)
fig,ax,cbar=plotzonalocn(latregrid, z_t, n2normz,
                    levels1=clevs,
                    levels2=clevs,
                    equalcontourscale=True,
                    figsize=(14,5),
                    title="N2 Scaling (max = %.1f/min = %.1f) "%(n2normz.max(),n2normz.min()))

# Add axis labels and plot title                                                                                                                                                                                              
plt.xlabel('Latitude')                                                                                                                                                                                                
plt.ylabel('Scaling factor')                                                                                                                                                                                              

# Customize y tick lables                                                                                                                                                                                                     
yticks=clevs                                                                                                                                                                                                          
cbar.set_ticks(yticks)                                                                                                                                                                                                
cbar.set_ticklabels(yticks)                                                                                                                                                                                           
cbar.set_label('fraction')                                                                                                                                                                                            

fig.savefig('Fig_31_N2_NORM_zonal_depth.png',bbox_inches='tight',dpi=200)

In [66]:

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Out[66]:

The alternate derivation of $\sigma_{vi}$:

$ R_i = {N^2 }$

$ N^2 = {{-g _0 }} $

After hydrostatic and geostrophic approximations

$f \frac {\partial v}{\partial z} = {{-g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad f \frac {\partial u}{\partial z} = {{g \over \rho_0 }\frac {\partial \rho}{\partial y}} $

so

$\frac {\partial v}{\partial z} = {{-1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad \frac {\partial u}{\partial z} = {{1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial y}} $

$\therefore$

$ R_i = {f^2 N^2 _}= $

$\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}$

$RX_1 = RX_{east} = \Delta\rho_x = \rho_{i+1,j} - \rho_{i,j}$

$RY_1 = RY_{north} = \Delta\rho_y = \rho_{i,j+1} - \rho_{i,j}$

$RZ_1 = RZ_{k+1} = \Delta\rho_z = \rho_{k} - \rho_{k+1}$

$\displaystyle{1 \over L_{R_i}} \displaystyle\int_{2000m}^{100m} \left\lbrace { {-g\over\rho_0}{\frac {\partial \rho} {\partial z}} \over { {g^2 \over \rho_0^2 } \left[( \frac{\partial \rho}{\partial y})^2 + ( \frac{\partial \rho}{\partial z})^2\right] } } \right\rbrace dz$

Note: missing $f^2$ which will be cancelled when forming $\sigma_{vi}$

$\quad\quad$ so $\cdots$ this is not $R_i$

Implementation notes

Numerator : Top $= -grav * RZ_{SAVE}(\cdots k+1) * dzwr(k)$

Denominator :

$

$work3 = {\left( TOP \over (grav^2*(work1+work2))\right)}*dzw(k)$

Notes:

1)Need to be careful at top and bottom of ocean
2)Accurate dzw(k) for each (i,j) to form $L_{R_i}$
3) When constructing $sigma$ itself, use $RZ_{SAVE}$ with a minimum N value
4) use eps2

POP Implementation of Mesoscale eddy diffusivity based on mixing theory

Implementation Notes on Diffusivity with Steering Level Suppression (Gokhan and John Marshall)

\(\quad\quad\quad\quad\quad\quad\) Eddy Length Scales:

\(\quad\quad\quad\quad\quad\quad\) Eddy Velocity:

\(\quad\quad\quad\quad\quad\quad\) Zonal phase speed:

References:

Questions:

\(\color{green}{\quad CESM \quad Implementation \quad of \quad Steering \quad Parameterization}\)

\(\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} \)

\(\quad\quad\quad\quad\quad\quad\quad\) First Baroclinic Wave Speed \(c_r\) CESM

\(\quad\quad\quad\quad\quad\quad\) Chelton Baroclinic gravity wave phase speed

\(\quad\quad\quad\quad\quad\quad\) First Baroclinic Rossby Radius CESM \(\quad L_{eddy}= min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}}).\)

\(\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})\)

\(\quad\quad\quad\quad\quad\) Eddy Phase speed CESM

\(\quad \quad \quad \quad \)Eddy Phase speed CESM \((c = max(- \beta * L_{eddy}^2,-20) )\)

\(\quad\quad\quad\quad\quad\quad\) Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09

\(\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})\)

\(\quad\quad \quad \quad\quad \) Zonal Velocity CESM

\(\color{green}{(U-c)\quad Term}\)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) Bates et al

\(\color{green}{(U-c)^2 \quad Term}\)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) Bates et al

\(\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma}\)

\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) CESM

\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) Bates et al

\(\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}\)

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity \((u_{rms})\quad\) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} \) Bates et al

\(\color{green}{u_{rms}^2 \quad Term} \quad\)

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity Squared \((u_{rms}^2)\quad\) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 \) Bates et al

\(\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}\)

\(\quad\quad\quad\quad\quad\quad\) \({(U-c)}^2 \over {u_{rms}^2} \quad\) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} \) Bates et al

\(\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}\)

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor CESM

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor Bates

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) CESM

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) Bates

\(\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}\)

\(\quad\quad\quad\quad\quad\quad\quad\) LMIX CESM

\(\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})\)

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) CESM

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) Bates

\(\quad\quad\quad\quad\quad\quad\quad\) (Zonal x Depth) CESM

Here is the Bate's version

\(\quad\quad\quad\quad\quad\quad\quad\) Zonal average of the N2 normalized scaling CESM

The alternate derivation of \(\sigma_{vi}\):

Implementation notes